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I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are analytic, for example the function given here. Moreover from this particular question asked on MSE we also know that the set of all Non-analytic smooth functions are dense in the space of continuous function on compact interval in $sup$-norm. So my questions are

Questions :

1) Suppose we have a smooth manifold. Does it always has an analytic structure ?

2) Like approximation of continuous functions by polynomials, does it make sense to say something similar about ${approximation}$ $by$ or $on$ manifolds ?

3) Suppose the answer to 2nd question is affirmative. Then since the set of all non-analytic smooth functions are dense as described above, can we say that any smooth manifold can be $approximated$ by manifolds having an analytic structure ?


I have thought about the 1st problem and I think that the answer is NO. But I wasn't able to produce a counter-example. I was not able to proceed by taking a non-analytic smooth function and to do something with that. As for 2) and 3) I don't have any idea about them.

Thanks in advance.

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  • $\begingroup$ 1) was asked on MO: mathoverflow.net/questions/8789/…. Apparently the answer is yes. $\endgroup$ – Qiaochu Yuan Feb 2 '15 at 7:16
  • $\begingroup$ @QiaochuYuan : Thanks a lot for that link. that is exactly what i was looking for, for 1). please let me know about any ideas you have for 2) and 2) also. $\endgroup$ – wanderer Feb 2 '15 at 7:35
  • $\begingroup$ You should take a look at Chapter 2 of Hirsch's "Differential Topology", in particular section 5. It provides references for some things you're looking for. $\endgroup$ – user98602 Feb 2 '15 at 7:44

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